Can a CofE minister allow a non-religious funeral to take place in her church?

A non-Christian funeral is not permitted in a church, but it is worth bearing in mind that a family looking for a totally non-Christian funeral cannot really achieve that in a church anyway - by its very nature, the building, the furnishings, the stained glass etc, all speak of the Christian faith. Apart from services which include formal worship, a church may be host to many non-religious community events like fetes, exercise classes, coffee mornings, etc, as a way of welcoming all. And there is no reason why a minister can't support any bereaved family in their parish, simply by being there for them, offering sympathy, prayer and a listening ear before and after, even if they don't have a Church of England funeral.